#412959

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,18)(7,19,22,13)(8,25,30,41)(9,31,34,43)(10,17,11,16)(14,29,42,28)(15,32,44,33)(20,47,49,39)(21,51,52,61)(23,37,24,38)(26,45,50,35)(27,57,60,40)(36,53,46,54)(48,65,67,62)(55,59,64,58)(56,63,68,66);(1,2)(3,4)(5,7,10,24,12,22,11,23)(6,13,16,38,18,19,17,37)(8,26,28,20,30,50,29,49)(9,21,32,54,34,52,33,53)(14,39,25,35,42,47,41,45)(15,36,43,51,44,46,31,61)(27,56,58,48,60,68,59,67)(40,63,55,62,57,66,64,65);(2,4)(5,8)(6,14)(7,20,22,49)(9,27)(10,29)(11,28)(12,30)(13,35,19,45)(15,40)(16,41)(17,25)(18,42)(21,48,52,67)(23,50,24,26)(31,55)(32,59)(33,58)(34,60)(36,62,46,65)(37,47,38,39)(43,64)(44,57)(51,63,61,66)(53,68,54,56);(5,9,10,32,12,34,11,33)(6,15,16,43,18,44,17,31)(7,21,24,54,22,52,23,53)(8,27,29,59,30,60,28,58)(13,36,38,51,19,46,37,61)(14,40,41,64,42,57,25,55)(20,48,26,56,49,67,50,68)(35,62,39,63,45,65,47,66)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320155);